src/HOL/Library/Wfrec.thy
 author Andreas Lochbihler Fri Sep 20 10:09:16 2013 +0200 (2013-09-20) changeset 53745 788730ab7da4 parent 44259 b922e91dd1d9 child 54482 a2874c8b3558 permissions -rw-r--r--
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```     1 (*  Title:      HOL/Library/Wfrec.thy
```
```     2     Author:     Tobias Nipkow
```
```     3     Author:     Lawrence C Paulson
```
```     4     Author:     Konrad Slind
```
```     5 *)
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```     6
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```     7 header {* Well-Founded Recursion Combinator *}
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```     8
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```     9 theory Wfrec
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```    10 imports Main
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```    11 begin
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```    12
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```    13 inductive
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```    14   wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b => bool"
```
```    15   for R :: "('a * 'a) set"
```
```    16   and F :: "('a => 'b) => 'a => 'b"
```
```    17 where
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```    18   wfrecI: "ALL z. (z, x) : R --> wfrec_rel R F z (g z) ==>
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```    19             wfrec_rel R F x (F g x)"
```
```    20
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```    21 definition
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```    22   cut        :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b" where
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```    23   "cut f r x == (%y. if (y,x):r then f y else undefined)"
```
```    24
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```    25 definition
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```    26   adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool" where
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```    27   "adm_wf R F == ALL f g x.
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```    28      (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"
```
```    29
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```    30 definition
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```    31   wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b" where
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```    32   "wfrec R F == %x. THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
```
```    33
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```    34 lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"
```
```    35 by (simp add: fun_eq_iff cut_def)
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```    36
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```    37 lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"
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```    38 by (simp add: cut_def)
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```    39
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```    40 text{*Inductive characterization of wfrec combinator; for details see:
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```    41 John Harrison, "Inductive definitions: automation and application"*}
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```    42
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```    43 lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. wfrec_rel R F x y"
```
```    44 apply (simp add: adm_wf_def)
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```    45 apply (erule_tac a=x in wf_induct)
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```    46 apply (rule ex1I)
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```    47 apply (rule_tac g = "%x. THE y. wfrec_rel R F x y" in wfrec_rel.wfrecI)
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```    48 apply (fast dest!: theI')
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```    49 apply (erule wfrec_rel.cases, simp)
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```    50 apply (erule allE, erule allE, erule allE, erule mp)
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```    51 apply (fast intro: the_equality [symmetric])
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```    52 done
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```    53
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```    54 lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"
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```    55 apply (simp add: adm_wf_def)
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```    56 apply (intro strip)
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```    57 apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)
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```    58 apply (rule refl)
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```    59 done
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```    60
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```    61 lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"
```
```    62 apply (simp add: wfrec_def)
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```    63 apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)
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```    64 apply (rule wfrec_rel.wfrecI)
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```    65 apply (intro strip)
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```    66 apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
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```    67 done
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```    68
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```    69
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```    70 text{** This form avoids giant explosions in proofs.  NOTE USE OF ==*}
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```    71 lemma def_wfrec: "[| f==wfrec r H;  wf(r) |] ==> f(a) = H (cut f r a) a"
```
```    72 apply auto
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```    73 apply (blast intro: wfrec)
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```    74 done
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```    75
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```    76
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```    77 subsection {* Nitpick setup *}
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```    78
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```    79 axiomatization wf_wfrec :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
```
```    80
```
```    81 definition wf_wfrec' :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
```
```    82 [nitpick_simp]: "wf_wfrec' R F x = F (cut (wf_wfrec R F) R x) x"
```
```    83
```
```    84 definition wfrec' ::  "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
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```    85 "wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x
```
```    86                 else THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
```
```    87
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```    88 setup {*
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```    89   Nitpick_HOL.register_ersatz_global
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```    90     [(@{const_name wf_wfrec}, @{const_name wf_wfrec'}),
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```    91      (@{const_name wfrec}, @{const_name wfrec'})]
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```    92 *}
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```    93
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```    94 hide_const (open) wf_wfrec wf_wfrec' wfrec'
```
```    95 hide_fact (open) wf_wfrec'_def wfrec'_def
```
```    96
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```    97 subsection {* Wellfoundedness of @{text same_fst} *}
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```    98
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```    99 definition
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```   100  same_fst :: "('a => bool) => ('a => ('b * 'b)set) => (('a*'b)*('a*'b))set"
```
```   101 where
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```   102     "same_fst P R == {((x',y'),(x,y)) . x'=x & P x & (y',y) : R x}"
```
```   103    --{*For @{text rec_def} declarations where the first n parameters
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```   104        stay unchanged in the recursive call. *}
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```   105
```
```   106 lemma same_fstI [intro!]:
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```   107      "[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R"
```
```   108 by (simp add: same_fst_def)
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```   109
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```   110 lemma wf_same_fst:
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```   111   assumes prem: "(!!x. P x ==> wf(R x))"
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```   112   shows "wf(same_fst P R)"
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```   113 apply (simp cong del: imp_cong add: wf_def same_fst_def)
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```   114 apply (intro strip)
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```   115 apply (rename_tac a b)
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```   116 apply (case_tac "wf (R a)")
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```   117  apply (erule_tac a = b in wf_induct, blast)
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```   118 apply (blast intro: prem)
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```   119 done
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```   120
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```   121 end
```